We verify the -finiteness of by covering each measurable set by countably many measurable rectangles with finite-measure sides. Since the sides’ measures are finite, the measure of the rectangle itself is the product of two finite numbers, and is thus finite.

We call the measure the “product” of the measures and , and we write . We thus have a -finite measure space that we call the “cartesian product” of the spaces and .

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I think that Feynman’s path integrals of quantum field theory might provide the kind of product measures that we were discussing although I am not so sure how rigorous they are considered to be mathemnatically. I wished to email this note to you, but I could not figure out where to find your email. It is probably not posted.

They’re completely unrigorous, mathematically; that’s the whole issue with them! The path-integral is a wonderful heuristic, that can be interpreted in many cases to give a mathematically-sensible statement.

But in general there is no known coherent analogue of something like Lebesgue measure on function spaces — no way to “integrate over all paths” as the plain reading of the path-integral formulation suggests. Finding a universally-applicable interpretation of the path-integral heuristic instead of problem-specific ad hoc methods is the single greatest puzzle of mathematical physics.

[…] of . Now if we have a measure on and Lebesgue measure on the Borel sets, we can define the product measure on . Since we know and are both measurable, we can investigate their measures. I assert […]

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